TFPT scientific source excerpt
Original repository path: experiments/theory-contracts/joint-native-algebra-selection-20261016/ORIGINAL-DENSITY-GEOMETRY-AND-TIME-HISTORIES.txt
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000138 | 4. Full state and native erasure response, computed directly in CAR
000139 | 
000140 | Use the same positive vectors U and source matrix V=M^T O U. Its entries
000141 | are Gaussian rationals. Native h_a(-n) maps to sum_k V_ka h_k(-n), where
000142 | 
000143 |  h_k(-n)=sum_(r=k mod8) :a_(r+8n)^* a_r:.
000144 | 
000145 | These bilinears reconstruct the previous full response
000146 | R_(b,4)f_i=[t^4]f_i(b+sum_(n>=1)h_+(-n)t^n), b=V^T delta.
000147 | They retain every partition4,31,22,211,1111. The independently computed
000148 | CAR Gram agrees entrywise with the previous complete bosonic Fock Gram.
000149 | Each state has charge0 and ORIGINAL source energy32, not source energy4.
000150 | 
000151 | J1 gives energy31, J-1 energy33. All eight forward and eight backward
000152 | cyclic root charge sectors are retained, including their sea signs.
000153 | Both response Grams have rank5, and the full VECTOR relation is
000154 | 
000155 |  (J1 J-1-J-1 J1)R_(b,4)f_i=R_(b,4)f_i.
000156 | 
000157 | Consequently G_+-G_-=B4. In BOTH original quarters G_- is not proportional
000158 | to B4: this current acts directionally even after accounting for the actual
000159 | input metric. The generalized polynomial det(G_--lambda B4) is squarefree
000160 | of degree5. All five eigenvalues are distinct; no degenerate 3+2 block is
000161 | supplied by this response. A further grouping would need justification.
000162 | 
000163 | The actual erasure a_a=h_(bar V_a)(1) still gives exactly
000164 | 
000165 |  a_a R_(b,4) f_i=R_(b,3) partial_a f_i.
000166 | 
000167 | Its commutator with the new original current is fixed without coefficients:
000168 | 
000169 |  [a_a,J_-1]=sum_r (bar V_(r+1,a)-bar V_(r,a)) a_(r-7)^* a_r.
000170 | 
000171 | All20 full source Ward outputs are nonzero, with every charged component
000172 | retained. This links the original density, its actual quartic response and
000173 | the existing derivative-erasure20. It is not a physical fermion/Yukawa map.
000174 | 
000175 | The complete original oscillator decomposition supplies a further identity:
000176 | 
000177 |  8K_alpha=Q(Q-1)/2+alpha Q+sum_(m>0) J_-m J_m.
000178 | 
000179 | For the actual neutral energy32 inputs this is the finite EXACT statement
000180 | sum_(m=1..32) Gram(J_m R_(b,4))=32 B4. Every one of the32 matrices is saved;
000181 | all m>32 annihilate by positive source energy. This is an energy bound, not
000182 | a chosen mode cutoff or discarded memory. Individual nonscalar responses
000183 | and the fixed total energy coexist.
000184 | 
000185 | The exact CAR calculation uses [-34,34]. Charge0 energy<=33 implies every
000186 | particle index<=32 and every hole index>=-33. All operator paths used here
000187 | stay below that energy. Every omitted edge bilinear annihilates the input;
000188 | signs are (-1) to the number of occupied modes strictly between its endpoints.
000189 | Thus the window implements these complete finite-energy responses exactly.
000190 | 
000191 | 5. Original geometric modes are not all folded canonical stress histories
000192 | 
000193 | The original one-channel fermion stress with its retained twist has
000194 | 
000195 |  Lcal_m^alpha=sum_r (r+m/2+alpha):a_r^* a_(r+m):
000196 |                      +(alpha-1/2)^2/2 delta_m0.
000197 | 
000198 | The constant is the original twisted conformal ground weight, while the
000199 | excitation generator is Lcal_0^alpha-(alpha-1/2)^2/2=8K_alpha.
000200 | This stress is also fixed by the same original density Sugawara identity;
000201 | it is not an independently chosen E8 improvement or clock.
000202 | 
000203 | The original cover/fold relation is
000204 | Lhat_n=Lcal_(8n)^alpha/8+21/64 delta_n0.
000205 | This is the ACTUAL twisted c8 geometric action. In the fixed current
000206 | dictionary its stress is Y_d(T)=T+d.h/z+||d||^2/(2z^2), not silently the
000207 | unshifted T on the old primary/vacuum seeds. Even for the folded geometric
000208 | subalgebra, the previous all-mode theorem cannot be extended to a different
000209 | operator/input pair merely by the same symbol 'stress'. Additionally, the
000210 | folded geometric c8 action samples only original indices divisible by8.
000211 | Original Lcal_+-1 changes fold energy by1/8 and produces charged cyclic D8
000212 | outputs. It is outside the earlier unshifted canonical c8-only class. The
000213 | earlier theorem is retained in its precise original source/input premises.
000214 | 
000215 | The full original CAR calculation verifies on every actual input vector
000216 | 
000217 |  [Lcal_1^alpha,J_-1]=(Q+alpha-1/2),
000218 |  [Lcal_1^alpha,Lcal_-1^alpha]=2Lcal_0^alpha.
000219 | 
000220 | The second RHS equals(64+(alpha-1/2)^2) on these neutral source energy32
000221 | inputs; omitting the twisted ground term fails this test.
000222 | 
000223 | Let G_L be the actual Lcal_1 response Gram. The two B4-selfadjoint response
000224 | operators B4^-1 G_- and B4^-1 G_L have common commutant dimension1, checked
000225 | over Gaussian rationals. Hence their generated complex *-algebra, with the
000226 | actual B4 adjoint, is M5, and they have no common reducing rank2/3 split.
000227 | This excludes a COMMON SPECTRAL-SECTOR identification for this response pair,
000228 | not a physical axiom requiring every mark to commute with all responses.
000229 | Nonlinear event readouts and other full source histories remain available.
000230 | 
000231 | In the saved COMPLETE current family, mode30 has rank2 and mode29 rank3.
000232 | This is not a derived Coxeter30 dictionary: J30 lowers original energy by30,
000233 | while a Coxeter clock is a specified unitary finite-order action. Moreover
000234 | the exact mode30 nullspace is none of the60 original polar rank3 kernels,
000235 | in BOTH quarters. The positive generalized overlap
000236 | tr((B4^-1 G29)(B4^-1 G30)) is nonzero, so these adjacent rank3/rank2
000237 | responses are not ORTHOGONALLY complementary in the actual B4 metric.
000238 | Their sum has rank5: they DO form an oblique algebraic direct sum. A positive
000239 | relative normalization can consequently build a rank2 projector for this
000240 | SUPPLIED mode pair, as in the previously proved general relative-response
000241 | theorem. This control does not exclude that construction or establish its
000242 | literal match to an original polar flag. No original rule selecting this
000243 | particular pair29/30 is supplied. They end in charge0 energies3 and2,
000244 | whose full source dimensions are p(3)=3 and p(2)=2. Dimension counts alone
000245 | do not identify them with the native carrier split. No mode30 selection
000246 | rule or eigenvalue grouping is adopted.
000247 |